9.1 Solvable groups and radical extensions

Solvable groups are the key to understanding which polynomials can be solved by radicals. They have a special structure that allows us to break them down into simpler pieces, just like we can break down radical solutions into simpler steps.

Radical extensions are fields we build by adding roots to our starting field. They're closely tied to solvable groups. If a polynomial's Galois group is solvable, we can solve it using radicals. This connection is at the heart of Galois theory.

Solvable Groups and Their Properties

Definition and Subnormal Series

• A group $G$ is solvable if it has a finite subnormal series $G = G₀ ⊇ G₁ ⊇ ... ⊇ Gₙ = \{e\}$ such that each factor group $Gᵢ/Gᵢ₊₁$ is abelian
• The subnormal series is a chain of subgroups, each normal in the previous one, ending with the trivial subgroup $\{e\}$
• The factor groups $Gᵢ/Gᵢ₊₁$ are the quotient groups formed by adjacent subgroups in the series

Properties and Examples

• Every abelian group is solvable, as the subnormal series can be chosen to have length 1 (the group itself and the trivial subgroup)
• The class of solvable groups is closed under taking subgroups, quotient groups, and extensions
  • If $G$ is solvable and $H ≤ G$, then $H$ is solvable
  • If $G$ is solvable and $N ◁ G$, then $G/N$ is solvable
  • If $N ◁ G$, $N$ and $G/N$ are solvable, then $G$ is solvable
• A finite group is solvable if and only if its composition series has factors that are all cyclic groups of prime order
• The symmetric group $S_n$ is solvable for $n ≤ 4$, but not for $n ≥ 5$ ($S_5$ is the smallest non-solvable symmetric group)
• Solvable groups play a crucial role in the study of solutions to polynomial equations by radicals

Solvable Groups vs Radical Extensions

Solvability by Radicals

• A polynomial $f(x) ∈ F[x]$ is solvable by radicals if its splitting field over $F$ can be obtained by successively adjoining roots of elements
• The roots of a polynomial solvable by radicals can be expressed using radicals, which are solutions to equations of the form $x^n = a$
• The Abel-Ruffini theorem states that for $n ≥ 5$, there exist polynomials of degree $n$ that are not solvable by radicals, as their Galois groups are not solvable

Galois Groups and Solvability

• The Galois group of a polynomial $f(x)$ over a field $F$ is solvable if and only if $f(x)$ is solvable by radicals over $F$
• If a polynomial has a solvable Galois group, then its roots can be expressed using radicals
• The solvability of a polynomial's Galois group determines whether the polynomial can be solved by radicals

Constructing Radical Extensions

Definition and Process

• A radical extension of a field $F$ is an extension field obtained by adjoining roots of elements of $F$
• To construct a radical extension for a polynomial $f(x) ∈ F[x]$:
  1. Factor $f(x)$ into irreducible factors over $F$
  2. For each irreducible factor $g(x)$, adjoin a root $α$ of $g(x)$ to $F$ to obtain a larger field $F(α)$
  3. Repeat the process of factoring and adjoining roots until $f(x)$ splits completely in the resulting extension field
• The splitting field of $f(x)$ over $F$ is the smallest field containing $F$ and all the roots of $f(x)$, and it is a radical extension if $f(x)$ is solvable by radicals

Examples

• Consider the polynomial $f(x) = x^3 - 2$ over $ℚ$
  • $f(x)$ is irreducible over $ℚ$, so adjoin a root $\sqrt[3]{2}$ to obtain $ℚ(\sqrt[3]{2})$
  • $f(x)$ splits completely in $ℚ(\sqrt[3]{2})$, so this is the splitting field and a radical extension
• For the polynomial $g(x) = x^4 - 4x^2 + 2$ over $ℚ$
  • Factor $g(x) = (x^2 - 2)^2 - 2 = ((x^2 - 2) - \sqrt{2})((x^2 - 2) + \sqrt{2})$
  • Adjoin $\sqrt{2}$ to $ℚ$ to obtain $ℚ(\sqrt{2})$
  • Factor the quadratic terms in $ℚ(\sqrt{2})$ and adjoin their roots to obtain the splitting field

Determining Solvability of Groups

Techniques and Criteria

• Check if the group is abelian. If so, it is solvable
• For a non-abelian group, try to find a subnormal series with abelian factor groups
• If the group is finite, examine its composition series and check if all factor groups are cyclic of prime order
• For the symmetric group $S_n$, it is solvable for $n ≤ 4$ and not solvable for $n ≥ 5$
• Use the closure properties of solvable groups:
  • If a group is built from solvable groups using extensions, subgroups, or quotients, then it is solvable
  • If a group has a non-solvable subgroup or quotient group, then the original group is not solvable

Examples

• The alternating group $A_4$ is solvable, as it has a subnormal series $A_4 ⊇ V_4 ⊇ \{e\}$ with abelian factor groups ($V_4$ is the Klein four-group)
• The dihedral group $D_8$ is solvable, as it has a subnormal series $D_8 ⊇ ⟨r^2, s⟩ ⊇ ⟨r^2⟩ ⊇ \{e\}$ with abelian factor groups ($r$ is rotation, $s$ is reflection)
• The quaternion group $Q_8$ is not solvable, as it has a non-abelian subgroup isomorphic to $Q_8$ itself
• The general linear group $\text{GL}_2(ℤ_3)$ is solvable, as it has a composition series with cyclic factor groups of prime order